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72-542: A definite integral computes the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line . Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an antiderivative , a function whose derivative is the given function; in this case, they are also called indefinite integrals . The fundamental theorem of calculus relates definite integration to differentiation and provides

144-424: A 2 − 1 b 2 ) + x y ( 1 a 2 + 1 b 2 ) {\displaystyle z=\left({\frac {x^{2}+y^{2}}{2}}\right)\left({\frac {1}{a^{2}}}-{\frac {1}{b^{2}}}\right)+xy\left({\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}\right)} and if a = b then this simplifies to z = 2 x y

216-530: A 2 + 4 v 2 b 2 a 2 b 2 ( 1 + 4 u 2 a 4 + 4 v 2 b 4 ) 3 {\displaystyle H(u,v)={\frac {a^{2}+b^{2}+{\frac {4u^{2}}{a^{2}}}+{\frac {4v^{2}}{b^{2}}}}{a^{2}b^{2}{\sqrt {\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{3}}}}}} which are both always positive, have their maximum at

288-418: A 2 . {\displaystyle z={\frac {2xy}{a^{2}}}.} Finally, letting a = √ 2 , we see that the hyperbolic paraboloid z = x 2 − y 2 2 . {\displaystyle z={\frac {x^{2}-y^{2}}{2}}.} is congruent to the surface z = x y {\displaystyle z=xy} which can be thought of as

360-425: A 2 . {\displaystyle z={\frac {y^{2}}{b^{2}}}-{\frac {x^{2}}{a^{2}}}.} In this position, the hyperbolic paraboloid opens downward along the x -axis and upward along the y -axis (that is, the parabola in the plane x = 0 opens upward and the parabola in the plane y = 0 opens downward). Any paraboloid (elliptic or hyperbolic) is a translation surface , as it can be generated by

432-396: A 2 b 2 ( 1 + 4 u 2 a 4 + 4 v 2 b 4 ) 2 {\displaystyle K(u,v)={\frac {-4}{a^{2}b^{2}\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{2}}}} and mean curvature H ( u , v ) = −

504-456: A 2 b 2 ( 1 + 4 u 2 a 4 + 4 v 2 b 4 ) 2 {\displaystyle K(u,v)={\frac {4}{a^{2}b^{2}\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{2}}}} and mean curvature H ( u , v ) = a 2 + b 2 + 4 u 2

576-573: A 2 + b 2 − 4 u 2 a 2 + 4 v 2 b 2 a 2 b 2 ( 1 + 4 u 2 a 4 + 4 v 2 b 4 ) 3 . {\displaystyle H(u,v)={\frac {-a^{2}+b^{2}-{\frac {4u^{2}}{a^{2}}}+{\frac {4v^{2}}{b^{2}}}}{a^{2}b^{2}{\sqrt {\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{3}}}}}.} If

648-471: A and b are constants that dictate the level of curvature in the xz and yz planes respectively. In this position, the elliptic paraboloid opens upward. A hyperbolic paraboloid (not to be confused with a hyperboloid ) is a doubly ruled surface shaped like a saddle . In a suitable coordinate system, a hyperbolic paraboloid can be represented by the equation z = y 2 b 2 − x 2

720-629: A closed and bounded interval [ a , b ] and can be generalized to other notions of integral (Lebesgue and Daniell). In this section, f is a real-valued Riemann-integrable function . The integral over an interval [ a , b ] is defined if a < b . This means that the upper and lower sums of the function f are evaluated on a partition a = x 0 ≤ x 1 ≤ . . . ≤ x n = b whose values x i are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating f within intervals [ x i  , x i  +1 ] where an interval with

792-714: A rectangular hyperbolic paraboloid , by analogy with rectangular hyperbolas . A plane section of a hyperbolic paraboloid with equation z = x 2 a 2 − y 2 b 2 {\displaystyle z={\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}} can be Saddle roofs are often hyperbolic paraboloids as they are easily constructed from straight sections of material. Some examples: The pencil of elliptic paraboloids z = x 2 + y 2 b 2 ,   b > 0 , {\displaystyle z=x^{2}+{\frac {y^{2}}{b^{2}}},\ b>0,} and

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864-401: A surface integral , the curve is replaced by a piece of a surface in three-dimensional space . The first documented systematic technique capable of determining integrals is the method of exhaustion of the ancient Greek astronomer Eudoxus and philosopher Democritus ( ca. 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of divisions for which

936-460: A , b ] is its width, b − a , so that the Lebesgue integral agrees with the (proper) Riemann integral when both exist. In more complicated cases, the sets being measured can be highly fragmented, with no continuity and no resemblance to intervals. Using the "partitioning the range of f " philosophy, the integral of a non-negative function f  : R → R should be the sum over t of

1008-411: A bounded interval, subsequently more general functions were considered—particularly in the context of Fourier analysis —to which Riemann's definition does not apply, and Lebesgue formulated a different definition of integral , founded in measure theory (a subfield of real analysis ). Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed. These approaches based on

1080-453: A certain class of "simple" functions, may be used to give an alternative definition of the integral. This is the approach of Daniell for the case of real-valued functions on a set X , generalized by Nicolas Bourbaki to functions with values in a locally compact topological vector space. See Hildebrandt 1953 for an axiomatic characterization of the integral. A number of general inequalities hold for Riemann-integrable functions defined on

1152-408: A common plane, but not to each other. Hence the hyperbolic paraboloid is a conoid . These properties characterize hyperbolic paraboloids and are used in one of the oldest definitions of hyperbolic paraboloids: a hyperbolic paraboloid is a surface that may be generated by a moving line that is parallel to a fixed plane and crosses two fixed skew lines . This property makes it simple to manufacture

1224-450: A connection between integration and differentiation . Barrow provided the first proof of the fundamental theorem of calculus . Wallis generalized Cavalieri's method, computing integrals of x to a general power, including negative powers and fractional powers. The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Leibniz and Newton . The theorem demonstrates

1296-565: A connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Leibniz and Newton developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions with continuous domains. This framework eventually became modern calculus , whose notation for integrals

1368-409: A curve as an infinite sum of rectangles of infinitesimal width. Bernhard Riemann later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into infinitesimally thin vertical slabs. In the early 20th century, Henri Lebesgue generalized Riemann's formulation by introducing what is now referred to as

1440-525: A degenerate interval, or a point , should be zero . One reason for the first convention is that the integrability of f on an interval [ a , b ] implies that f is integrable on any subinterval [ c , d ] , but in particular integrals have the property that if c is any element of [ a , b ] , then: Signed area Too Many Requests If you report this error to the Wikimedia System Administrators, please include

1512-518: A function f over the interval [ a , b ] is equal to S if: When the chosen tags are the maximum (respectively, minimum) value of the function in each interval, the Riemann sum becomes an upper (respectively, lower) Darboux sum , suggesting the close connection between the Riemann integral and the Darboux integral . It is often of interest, both in theory and applications, to be able to pass to

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1584-458: A function f with respect to such a tagged partition is defined as thus each term of the sum is the area of a rectangle with height equal to the function value at the chosen point of the given sub-interval, and width the same as the width of sub-interval, Δ i = x i − x i −1 . The mesh of such a tagged partition is the width of the largest sub-interval formed by the partition, max i =1... n Δ i . The Riemann integral of

1656-401: A higher index lies to the right of one with a lower index. The values a and b , the end-points of the interval , are called the limits of integration of f . Integrals can also be defined if a > b : With a = b , this implies: The first convention is necessary in consideration of taking integrals over subintervals of [ a , b ] ; the second says that an integral taken over

1728-423: A hyperbolic paraboloid from a variety of materials and for a variety of purposes, from concrete roofs to snack foods. In particular, Pringles fried snacks resemble a truncated hyperbolic paraboloid. A hyperbolic paraboloid is a saddle surface , as its Gauss curvature is negative at every point. Therefore, although it is a ruled surface, it is not developable . From the point of view of projective geometry ,

1800-430: A hyperbolic paraboloid is one-sheet hyperboloid that is tangent to the plane at infinity . A hyperbolic paraboloid of equation z = a x y {\displaystyle z=axy} or z = a 2 ( x 2 − y 2 ) {\displaystyle z={\tfrac {a}{2}}(x^{2}-y^{2})} (this is the same up to a rotation of axes ) may be called

1872-413: A letter to Paul Montel : I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay

1944-405: A method to compute the definite integral of a function when its antiderivative is known; differentiation and integration are inverse operations. Although methods of calculating areas and volumes dated from ancient Greek mathematics , the principles of integration were formulated independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, who thought of the area under

2016-432: A moving parabola directed by a second parabola. In a suitable Cartesian coordinate system , an elliptic paraboloid has the equation z = x 2 a 2 + y 2 b 2 . {\displaystyle z={\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}.} If a = b , an elliptic paraboloid is a circular paraboloid or paraboloid of revolution . It

2088-404: A similar property of symmetry. Every plane section of a paraboloid made by a plane parallel to the axis of symmetry is a parabola. The paraboloid is hyperbolic if every other plane section is either a hyperbola , or two crossing lines (in the case of a section by a tangent plane). The paraboloid is elliptic if every other nonempty plane section is either an ellipse , or a single point (in

2160-490: A suitable class of functions (the measurable functions ) this defines the Lebesgue integral. A general measurable function f is Lebesgue-integrable if the sum of the absolute values of the areas of the regions between the graph of f and the x -axis is finite: In that case, the integral is, as in the Riemannian case, the difference between the area above the x -axis and the area below the x -axis: where Although

2232-457: A symmetrical paraboloidal dish are related by the equation 4 F D = R 2 , {\displaystyle 4FD=R^{2},} where F is the focal length, D is the depth of the dish (measured along the axis of symmetry from the vertex to the plane of the rim), and R is the radius of the rim. They must all be in the same unit of length . If two of these three lengths are known, this equation can be used to calculate

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2304-486: Is a parabolic cylinder (see image). The elliptic paraboloid, parametrized simply as σ → ( u , v ) = ( u , v , u 2 a 2 + v 2 b 2 ) {\displaystyle {\vec {\sigma }}(u,v)=\left(u,v,{\frac {u^{2}}{a^{2}}}+{\frac {v^{2}}{b^{2}}}\right)} has Gaussian curvature K ( u , v ) = 4

2376-425: Is a surface of revolution obtained by revolving a parabola around its axis. A circular paraboloid contains circles. This is also true in the general case (see Circular section ). From the point of view of projective geometry , an elliptic paraboloid is an ellipsoid that is tangent to the plane at infinity . The plane sections of an elliptic paraboloid can be: On the axis of a circular paraboloid, there

2448-413: Is a point called the focus (or focal point ), such that, if the paraboloid is a mirror, light (or other waves) from a point source at the focus is reflected into a parallel beam, parallel to the axis of the paraboloid. This also works the other way around: a parallel beam of light that is parallel to the axis of the paraboloid is concentrated at the focal point. For a proof, see Parabola § Proof of

2520-429: Is closed under taking linear combinations , and the integral of a linear combination is the linear combination of the integrals: Similarly, the set of real -valued Lebesgue-integrable functions on a given measure space E with measure μ is closed under taking linear combinations and hence form a vector space, and the Lebesgue integral is a linear functional on this vector space, so that: More generally, consider

2592-417: Is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. A tagged partition of a closed interval [ a , b ] on the real line is a finite sequence This partitions the interval [ a , b ] into n sub-intervals [ x i −1 , x i ] indexed by i , each of which is "tagged" with a specific point t i ∈ [ x i −1 , x i ] . A Riemann sum of

2664-505: Is drawn directly from the work of Leibniz. While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigour . Bishop Berkeley memorably attacked the vanishing increments used by Newton, calling them " ghosts of departed quantities ". Calculus acquired a firmer footing with the development of limits . Integration was first rigorously formalized, using limits, by Riemann . Although all bounded piecewise continuous functions are Riemann-integrable on

2736-501: Is not uncommon to leave out dx when only the simple Riemann integral is being used, or the exact type of integral is immaterial. For instance, one might write ∫ a b ( c 1 f + c 2 g ) = c 1 ∫ a b f + c 2 ∫ a b g {\textstyle \int _{a}^{b}(c_{1}f+c_{2}g)=c_{1}\int _{a}^{b}f+c_{2}\int _{a}^{b}g} to express

2808-448: Is of great importance to have a definition of the integral that allows a wider class of functions to be integrated. Such an integral is the Lebesgue integral, that exploits the following fact to enlarge the class of integrable functions: if the values of a function are rearranged over the domain, the integral of a function should remain the same. Thus Henri Lebesgue introduced the integral bearing his name, explaining this integral thus in

2880-405: Is shaped like an oval cup and has a maximum or minimum point when its axis is vertical. In a suitable coordinate system with three axes x , y , and z , it can be represented by the equation z = x 2 a 2 + y 2 b 2 . {\displaystyle z={\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}.} where

2952-491: The Lebesgue integral ; it is more general than Riemann's in the sense that a wider class of functions are Lebesgue-integrable. Integrals may be generalized depending on the type of the function as well as the domain over which the integration is performed. For example, a line integral is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting two points in space. In

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3024-454: The analytic function f ( z ) = z 2 2 = f ( x + y i ) = z 1 ( x , y ) + i z 2 ( x , y ) {\displaystyle f(z)={\frac {z^{2}}{2}}=f(x+yi)=z_{1}(x,y)+iz_{2}(x,y)} which is the analytic continuation of the R → R parabolic function f ( x ) = ⁠ x / 2 ⁠ . The dimensions of

3096-429: The differential of the variable x , indicates that the variable of integration is x . The function f ( x ) is called the integrand, the points a and b are called the limits (or bounds) of integration, and the integral is said to be over the interval [ a , b ] , called the interval of integration. A function is said to be integrable if its integral over its domain is finite. If limits are specified,

3168-435: The Riemann and Lebesgue integrals are the most widely used definitions of the integral, a number of others exist, including: The collection of Riemann-integrable functions on a closed interval [ a , b ] forms a vector space under the operations of pointwise addition and multiplication by a scalar, and the operation of integration is a linear functional on this vector space. Thus, the collection of integrable functions

3240-420: The area or volume was known. This method was further developed and employed by Archimedes in the 3rd century BC and used to calculate the area of a circle , the surface area and volume of a sphere , area of an ellipse , the area under a parabola , the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral . A similar method

3312-440: The areas between a thin horizontal strip between y = t and y = t + dt . This area is just μ { x  : f ( x ) > t }  dt . Let f ( t ) = μ { x  : f ( x ) > t } . The Lebesgue integral of f is then defined by where the integral on the right is an ordinary improper Riemann integral ( f is a strictly decreasing positive function, and therefore has a well-defined improper Riemann integral). For

3384-479: The box notation was difficult for printers to reproduce, so these notations were not widely adopted. The term was first printed in Latin by Jacob Bernoulli in 1690: "Ergo et horum Integralia aequantur". In general, the integral of a real-valued function f ( x ) with respect to a real variable x on an interval [ a , b ] is written as The integral sign ∫ represents integration. The symbol dx , called

3456-442: The case of a section by a tangent plane). A paraboloid is either elliptic or hyperbolic. Equivalently, a paraboloid may be defined as a quadric surface that is not a cylinder , and has an implicit equation whose part of degree two may be factored over the complex numbers into two different linear factors. The paraboloid is hyperbolic if the factors are real; elliptic if the factors are complex conjugate . An elliptic paraboloid

3528-545: The definite integral, with limits above and below the integral sign, was first used by Joseph Fourier in Mémoires of the French Academy around 1819–1820, reprinted in his book of 1822. Isaac Newton used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. The vertical bar was easily confused with . x or x ′ , which are used to indicate differentiation, and

3600-424: The details below. Request from 172.68.168.132 via cp1112 cp1112, Varnish XID 946188140 Upstream caches: cp1112 int Error: 429, Too Many Requests at Thu, 28 Nov 2024 08:37:40 GMT Paraboloid In geometry , a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry . The term "paraboloid" is derived from parabola , which refers to a conic section that has

3672-406: The dish, measured along the surface, is then given by R Q P + P ln ⁡ ( R + Q P ) , {\displaystyle {\frac {RQ}{P}}+P\ln \left({\frac {R+Q}{P}}\right),} where ln x means the natural logarithm of x , i.e. its logarithm to base e . The volume of the dish, the amount of liquid it could hold if

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3744-458: The foundations of modern calculus, with Cavalieri computing the integrals of x up to degree n = 9 in Cavalieri's quadrature formula . The case n = −1 required the invention of a function , the hyperbolic logarithm , achieved by quadrature of the hyperbola in 1647. Further steps were made in the early 17th century by Barrow and Torricelli , who provided the first hints of

3816-496: The geometric representation (a three-dimensional nomograph , as it were) of a multiplication table . The two paraboloidal R → R functions z 1 ( x , y ) = x 2 − y 2 2 {\displaystyle z_{1}(x,y)={\frac {x^{2}-y^{2}}{2}}} and z 2 ( x , y ) = x y {\displaystyle z_{2}(x,y)=xy} are harmonic conjugates , and together form

3888-471: The hyperbolic paraboloid z = x 2 a 2 − y 2 b 2 {\displaystyle z={\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}} is rotated by an angle of ⁠ π / 4 ⁠ in the + z direction (according to the right hand rule ), the result is the surface z = ( x 2 + y 2 2 ) ( 1

3960-533: The integral is called a definite integral. When the limits are omitted, as in the integral is called an indefinite integral, which represents a class of functions (the antiderivative ) whose derivative is the integrand. The fundamental theorem of calculus relates the evaluation of definite integrals to indefinite integrals. There are several extensions of the notation for integrals to encompass integration on unbounded domains and/or in multiple dimensions (see later sections of this article). In advanced settings, it

4032-429: The limit under the integral. For instance, a sequence of functions can frequently be constructed that approximate, in a suitable sense, the solution to a problem. Then the integral of the solution function should be the limit of the integrals of the approximations. However, many functions that can be obtained as limits are not Riemann-integrable, and so such limit theorems do not hold with the Riemann integral. Therefore, it

4104-434: The linearity holds for the subspace of functions whose integral is an element of V (i.e. "finite"). The most important special cases arise when K is R , C , or a finite extension of the field Q p of p-adic numbers , and V is a finite-dimensional vector space over K , and when K = C and V is a complex Hilbert space . Linearity, together with some natural continuity properties and normalization for

4176-517: The linearity of the integral, a property shared by the Riemann integral and all generalizations thereof. Integrals appear in many practical situations. For instance, from the length, width and depth of a swimming pool which is rectangular with a flat bottom, one can determine the volume of water it can contain, the area of its surface, and the length of its edge. But if it is oval with a rounded bottom, integrals are required to find exact and rigorous values for these quantities. In each case, one may divide

4248-685: The number of pieces increases to infinity, it will reach a limit which is the exact value of the area sought (in this case, 2/3 ). One writes which means 2/3 is the result of a weighted sum of function values, √ x , multiplied by the infinitesimal step widths, denoted by dx , on the interval [0, 1] . There are many ways of formally defining an integral, not all of which are equivalent. The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but are also occasionally for pedagogical reasons. The most commonly used definitions are Riemann integrals and Lebesgue integrals. The Riemann integral

4320-647: The origin, become smaller as a point on the surface moves further away from the origin, and tend asymptotically to zero as the said point moves infinitely away from the origin. The hyperbolic paraboloid, when parametrized as σ → ( u , v ) = ( u , v , u 2 a 2 − v 2 b 2 ) {\displaystyle {\vec {\sigma }}(u,v)=\left(u,v,{\frac {u^{2}}{a^{2}}}-{\frac {v^{2}}{b^{2}}}\right)} has Gaussian curvature K ( u , v ) = − 4

4392-442: The pencil of hyperbolic paraboloids z = x 2 − y 2 b 2 ,   b > 0 , {\displaystyle z=x^{2}-{\frac {y^{2}}{b^{2}}},\ b>0,} approach the same surface z = x 2 {\displaystyle z=x^{2}} for b → ∞ {\displaystyle b\rightarrow \infty } , which

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4464-476: The real number system are the ones most common today, but alternative approaches exist, such as a definition of integral as the standard part of an infinite Riemann sum, based on the hyperreal number system. The notation for the indefinite integral was introduced by Gottfried Wilhelm Leibniz in 1675. He adapted the integral symbol , ∫ , from the letter ſ ( long s ), standing for summa (written as ſumma ; Latin for "sum" or "total"). The modern notation for

4536-474: The reflective property . Therefore, the shape of a circular paraboloid is widely used in astronomy for parabolic reflectors and parabolic antennas. The surface of a rotating liquid is also a circular paraboloid. This is used in liquid-mirror telescopes and in making solid telescope mirrors (see rotating furnace ). The hyperbolic paraboloid is a doubly ruled surface : it contains two families of mutually skew lines . The lines in each family are parallel to

4608-416: The results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid . The next significant advances in integral calculus did not begin to appear until the 17th century. At this time, the work of Cavalieri with his method of indivisibles , and work by Fermat , began to lay

4680-409: The right end height of each piece (thus √ 0 , √ 1/5 , √ 2/5 , ..., √ 1 ) and sum their areas to get the approximation which is larger than the exact value. Alternatively, when replacing these subintervals by ones with the left end height of each piece, the approximation one gets is too low: with twelve such subintervals the approximated area is only 0.6203. However, when

4752-430: The rim were horizontal and the vertex at the bottom (e.g. the capacity of a paraboloidal wok ), is given by π 2 R 2 D , {\displaystyle {\frac {\pi }{2}}R^{2}D,} where the symbols are defined as above. This can be compared with the formulae for the volumes of a cylinder ( π R D ), a hemisphere ( ⁠ 2π / 3 ⁠ R D , where D = R ), and

4824-440: The several heaps one after the other to the creditor. This is my integral. As Folland puts it, "To compute the Riemann integral of f , one partitions the domain [ a , b ] into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of f ". The definition of the Lebesgue integral thus begins with a measure , μ. In the simplest case, the Lebesgue measure μ ( A ) of an interval A = [

4896-421: The sought quantity into infinitely many infinitesimal pieces, then sum the pieces to achieve an accurate approximation. As another example, to find the area of the region bounded by the graph of the function f ( x ) = x {\textstyle {\sqrt {x}}} between x = 0 and x = 1 , one can divide the interval into five pieces ( 0, 1/5, 2/5, ..., 1 ), then construct rectangles using

4968-451: The sum of fourth powers . Alhazen determined the equations to calculate the area enclosed by the curve represented by y = x k {\displaystyle y=x^{k}} (which translates to the integral ∫ x k d x {\displaystyle \int x^{k}\,dx} in contemporary notation), for any given non-negative integer value of k {\displaystyle k} . He used

5040-536: The third. A more complex calculation is needed to find the diameter of the dish measured along its surface . This is sometimes called the "linear diameter", and equals the diameter of a flat, circular sheet of material, usually metal, which is the right size to be cut and bent to make the dish. Two intermediate results are useful in the calculation: P = 2 F (or the equivalent: P = ⁠ R / 2 D ⁠ ) and Q = √ P + R , where F , D , and R are defined as above. The diameter of

5112-413: The vector space of all measurable functions on a measure space ( E , μ ) , taking values in a locally compact complete topological vector space V over a locally compact topological field K , f  : E → V . Then one may define an abstract integration map assigning to each function f an element of V or the symbol ∞ , that is compatible with linear combinations. In this situation,

5184-519: Was independently developed in China around the 3rd century AD by Liu Hui , who used it to find the area of the circle. This method was later used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere. In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen ( c.  965  – c.  1040  AD) derived a formula for

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